A survey on frame representations via dynamical sampling

TL;DR

This survey explores various methods of representing frames in Hilbert spaces through dynamical sampling, including orbit-based and subset-based approaches, highlighting the flexibility and limitations of bounded operators.

Contribution

It provides a comprehensive overview of existing results and discusses alternative frame representations using operator orbits and subsets, expanding the scope of dynamical sampling techniques.

Findings

Boundedness of operator T imposes restrictions on frame representations.

Allowing subset-based representations increases the class of representable frames.

Approximate representations enable greater flexibility in frame analysis.

Abstract

Dynamical sampling deals with representations of a frame $\{ f_k \}_{k=1}^\infty$ as an orbit $\{ T^n \varphi \}_{n=0}^\infty$ of a linear and possibly bounded operator $T$ acting on the underlying Hilbert space. It is known that the desire of boundedness of the operator $T$ puts severe restrictions on the frame $\{ f_k \}_{k=1}^\infty$. The purpose of the paper is to present an overview of the results in the literature and also discuss various alternative ways of representing a frame; in particular the class of considered frames can be enlarged drastically by allowing representations using only a subset $\{ T^{\alpha(k)} \varphi\}^\infty_{k=1}$ of the operator orbit $\{ T^n \varphi \}_{n=0}^\infty$. In general it is difficult to specify appropriate values for the scalars $\alpha(k)$ and the vector $\varphi;$ however, by accepting an arbitrarily small and controllable deviation between the given frame $\{ f_k \}_{k=1}^\infty$ and $\{ T^{\alpha(k)} \varphi\}_{k=1}^\infty$ we will be able to do so.